On the (in)finiteness of the image of Reshetikhin-Turaev representations

TL;DR

This paper provides a simple criterion to determine when Reshetikhin-Turaev representations of surface mapping class groups have infinite images, and applies it to specific cases, offering an alternative proof of existing results.

Contribution

It introduces a new criterion for infiniteness of Reshetikhin-Turaev representations and applies it to particular cases of punctured tori, extending previous work.

Findings

Criterion successfully determines infiniteness of representations.

Applied criterion to specific punctured tori cases.

Provided an alternative proof of Funar's result.

Abstract

We state a simple criterion to prove the infiniteness of the image of Reshetikhin-Turaev irreducible representations of the mapping class groups of surfaces. We use it to study some of the Reshetikhin-Turaev representations associated to the tori with one and two punctures and derive an alternative proof of a result of Funar.